As mentioned before, to uniquely decode \(\mathcal {C}_{\ell }\)  we will
inductively employ the list decoding machinery for expander walks
from [1]}. The list decoding algorithm can decode a direct
sum lifting \(\mathcal {C}^{\prime } = \operatorname{dsum}_{W(m)}(\mathcal {C})\)  as long as the graph \(G\)  is
sufficiently expanding, the walk length \(m-1\)  is large enough, and the
base code \(\mathcal {C}\)  has an efficient unique decoding algorithm
(see theo:listdechammer for details).
